Solution of the Linear Ordering Problem (NP= P)

arXiv:cs/0303008v1 [cs.CC] 14 Mar 2003

Solution of the Linear Ordering Problem (NP=P) G. Bolotashvili Email: [email protected]

We consider the following problem max

n n X X

cij xij

i=1,i6=j j=1

s. t.

0 6 xij 6 1, xij + xji = 1, 0 6 xij + xjk − xik 6 1, i 6= j, i 6= k, j 6= k, i, j, k = 1, ..., n.

We denote the corresponding polytope by Bn . The polytope Bn has integer vertices corresponding to feasible solutions of the linear ordering problem as well as non-integer vertices. We denote the polytope of integer vertices as Pn . Let us give an example of non-integer vertex in Bn and describe an exact facet cut. In what follows we will interested only in generating exact facet cuts. Fig. 1 shows a graph interpretation of a non-integer vertex [1], i1 •_?jT?TTTT

i2 ? gOOO • O

• j1

• j2

···

im

47 jjo• ?? TTTT OOO jojojojoo j j j T ??  TTTT O j o TTTTjjOjOjOjOjOojoooo ?? T j o O ?  T j o O  ?? jjjjj ooToToTTTTOTOOO TTTOTOO  jjj?j?j?j oo  TTOTOTO ? ooooo jjjjj

···

Fig. 1 1

• jm

where {i1 , ..., im }, {j1 , ..., jm } ⊂ {1, ..., n}; {i1 , ..., im } ∩ {j1 , ..., jm } = ∅; xil jq = 0, xjq il = 1, l 6= q, l, q = 1, ..., m; the other variables that are not shown at the Figure 1 are equal to 12 . This is the simplest non-integer vertex of the polytope Bn . For this vertex all adjacent integer vertices can be written as: where αk is any ordering from the set {j1 , ..., jm }\{jk }, βk is any ordering from the set {i1 , ..., im }\{ik },

αk ik jk βk ,

αkp ik jk ip jp βkp , where αkp is any ordering from the set {j1 , ..., jm }\{jk , jp }, βkp is any ordering from the set {i1 , ..., im }\{ik , ip }, k 6= p, k, p = 1, ..., m. All adjacent integer vertices, which number is equal to m [(m − 1)!]2 +

m(m − 1) [(m − 2)!]2 2

lie in one hyperplane f (x) = 2

m X

xil jl −

m X m X

xil jq = 1.

l=1 q=1

l=1

This hyperplane for f (x) ≤ 1 is a facet of the polytope Pn . Our aim is to determine exact facet cuts for any non-integer vertex of Bn (and not only for them) in an analogous fashion. Figures 2 and 3 show non-integer vertices of the polytope Bn : 1 •X1hQfLaC1CLQLQQ

• 4

2

3

4 •X1lfL1LLL

5

6= F = aCC m8• {• CC mrmr{mr{r{

11CCLCLQLQQQ {{{ m m r CCmmmrr {{

11 CCLLLQL{Q{QQ mC r { 11 CC{C{{LLLQQQmQmmmrmrrrCC{C{{

m Q L r C { Q 11{ CmCmmmLLrLrr QQQ{{ CC C {1 mm C r LL {{ QQ C { {m{mmm1m1 rrrrCCrC {{{LLLL Q QQQCQCC { Q r m{m / { 2/

8F r2• 11 LLL rrrrr

11 •r

11

11 3

4 •jVVVVVV hh• VVVV 111

hhhhhhh 1 2 VVV hhhh •

Fig. 2

Fig. 3

• 7

• 5

• 8

• 6

6

Noninteger vertex at Figure 2 has an oriented chain 7583 of the length 3, and non-integer vertex at Figure 3 has an oriented 614 chain of the length 2

2. The oriented chain 758 at Figure 2 is independent, that is if we exchange the chain 758 with any other chain the rest of the graph does not change, while the chains 81 at Figure 2 and 614 at Figure 3 are dependent. We define corresponding dependent and independent oriented chains. The following Theorems take place. Theorem 1 Let x0 be a noninteger vertex in Bn and assume that in graph interpretation there is a graph vertex i which is the begin or the end of all adjacent arcs. Assume that the vertex i can be repeated arbitrarily many times such that each of the new vertices has the same location with the other part of the graph as the vertex i. Then the new noninteger vertex, corresponding to the new graph, is a noninteger vertex of Bn , and in the new graph the vertices i and new inserted vertices may be put in any order. Theorem 2 Let x0 be a noninteger vertex in Bn . Then there does not exist corresponding dependent oriented chains of the length 4. The polytope Bn has noninteger vertices whose fractional components are equal to rl , r > 2, l < r, as well. For r = 2 after matrix transformation we get in all cases the following non unimodular minimal standard matrix:   1 −1 0  1 0 −1  . 0 1 1 For r = 3 after matrix transformation we get in all cases the known combination of two minimal standard matrices:   1 −1 0 −1 0  1 0 −1 0 0    .  0 1 1 0 0    1 0 0 0 −1  0 0 0 1 1 For any r after matrix transformation we get known combination of r − 1 minimal standard matrices. 3

Theorem 3 Let x0 be a noninteger vertex of Bn , which has fractional components rl11 , l1 < r1 , r1 > 3. Then we pass to an adjacent noninteger vertex with fractional components rl22 , r2 < r1 , l2 < r2 , by changing an equality in a basis (thus changing one or more minimal standard matrices). Let I1 , I2 , ..., Is ⊂ {1, ..., n}, and assume that each set Ip , p = 1, ..., s, corresponds to noninteger components of a vertex. Then for each Ip , p = 1, ..., s, we can construct a facet cut. If s = 1 a noninteger vertex is called simple. A noninteger vertex is called complex if s > 2. Thus we have given a general description of the polytope Bn . Theorem 4 Let x0 be a simple noninteger vertex of the polytope Bn with fractional components 12 , assume further that there does not exist dependent oriented chains with the length 3. Then all adjacent integer vertices lie in one hyperplane, this hyperplane is a facet of the polytope Pn , and it can be constructed by a polynomial algorithm. Now we describe the principle for constructing facets. Consider a noninteger vertex x0 . It can be defined as the solution of the following system of the linear basic equalities xil jl = 0, l = 1, ..., q;

(1) 2

xil jl + xjl kl − xil kl = 0, l = q + 1, ...,

n −n . 2

We introduce artificial variables xjl n+1 = 0, xil n+1 = 0, into the first q equalities of the system (1): xil jl + xjl n+1 − xil n+1 = 0, l = 1, ..., q. With the help of the notation xil jl + xjl kl − xil kl := x(il , jl , kl ), l = 1, ..., we rewrite the system (1) as follows: x(il , jl , kl ) = 0, l = 1, ..., 4

n2 − n . 2

n2 − n , 2

We can determine

n2 −n 2

linear independent adjacent integer vertices

n2 − n , x (is , js , ks ) = δ (is , js , ks ), s, q = 1, ..., 2 q

q

where δ q is either 1 or 0. We can prove that all adjacent integer vertices lie in the hyperplane: x(i1 , j1 , k1 ) . . . x(im , jm , km ) 1 1 δ (i1 , j1 , k1 ) . . . δ 1 (im , jm , km ) 1 =0 f (x) = ... m δ (i1 , j1 , k1 ) . . . δ m (im , jm , km ) 1 where m =

n2 −n . 2

Theorem 5 Let x0 be a simple noninteger vertex of the polytope Bn with fractional components 12 , assume further that there exist τ dependent oriented chains with the length 3. Then all adjacent integer vertices lie in 2τ hyperplanes, each of them is a facet of the polytope Pn , and can be constructed by a polynomial algorithm. Theorem 6 Let x0 be a simple noninteger vertex of the polytope Bn with fractional components rl , r > 3, l < r. Then we can construct all minimal standard matrices and corresponding noninteger vertices with fractional components 21 . For every such vertex we can construct facet cuts. Theorem 7 Let x0 be a complex noninteger vertex of the polytope Bn , and I1 , I2 , ..., Is correspond to noninteger values. Then we can construct facet cuts for each set Ip , p = 1, ..., s. Assume we have generated facet cuts. Solving the problem again we get the noninteger vertex x1 of the polytope 0 6 xij 6 1, xij + xji = 1, 0 6 xij + xjk − xik 6 1, i 6= j, i 6= k, j 6= k, i, j, k = 1, ..., n, fs1 6 fs (x) 6 fs2 , s = 1, ..., q.

5

Without loss of generality we may assume that the noninteger vertex x1 satisfies the following linear system: x(is , js , ks ) = 1, i = 1, ..., p, fs (x) = fs2 , s = 1, ..., q. Now we find all adjacent integer vertices. If all of them lie in one hyperplane and this hyperplane is a facet of Pn then we generate this facet and re-solve the problem with a new facet. In case we cannot determine the facet we solve the auxiliary problem: P Pq fs (x)−fs1  p max x(i , j , k ) + , s s s s=1 s=1 fs2 −fs1 0 6 xij 6 1, xij + xji = 1, 0 6 xij + xjk − xik 6 1, i 6= j, i 6= k, j 6= k, i, j, k = 1, ..., n, With the solution of this problem we can construct the facet of the polytope Pn . In the case of theorems 5 and 6 we can construct the necessary facets by means of a polynomial algorithm.

References [1] Bolotashvili G., Kovalev M., Girlich E. New Facets of the linear ordering Polytope. SIAM J. Discrete Mathematics 12(3):326-336, 1999.

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